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Machine Learning for Fluid Mechanics

Steven Brunton, Bernd Noack, Petros Koumoutsakos

arXiv:1905.11075v3physics.flu-dyncs.LGstat.ML

TL;DR

Fluid mechanics faces high-dimensional, nonlinear problems alongside rapidly growing data, creating a need to understand how machine learning can support analysis and engineering tasks. This review synthesizes machine-learning methods and applications in fluid modeling, optimization, and control, concluding that these approaches can enrich fluid-mechanics research while requiring stronger guarantees and integration of known physics.

  • Problem

    High-dimensionality and nonlinearity limit closed-form analysis and real-time optimization and control, while growing fluid-mechanics data motivates machine-learning approaches.

  • Method

    The review surveys machine-learning methodologies and their applications to fluid-flow understanding, modeling, optimization, and control across experiments and simulations.

  • Results

    The review highlights machine-learning successes in dimensionality reduction, feature extraction, PIV processing, super-resolution, reduced-order modeling, turbulence closure, shape optimization, and flow control.

  • Takeaways & Limitations

    Machine learning offers a powerful framework relevant to modeling, optimization, and control of fluid flows, especially when combined with fluid-mechanics expertise.

  • Takeaways & Limitations

    Machine-learning algorithms often lack guarantees for performance, robustness, or convergence, and incorporating known flow physics remains challenging.

Abstract

from arXiv · show

The field of fluid mechanics is rapidly advancing, driven by unprecedented volumes of data from field measurements, experiments and large-scale simulations at multiple spatiotemporal scales. Machine learning offers a wealth of techniques to extract information from data that could be translated into knowledge about the underlying fluid mechanics. Moreover, machine learning algorithms can augment domain knowledge and automate tasks related to flow control and optimization. This article presents an overview of past history, current developments, and emerging opportunities of machine learning for fluid mechanics. It outlines fundamental machine learning methodologies and discusses their uses for understanding, modeling, optimizing, and controlling fluid flows. The strengths and limitations of these methods are addressed from the perspective of scientific inquiry that considers data as an inherent part of modeling, experimentation, and simulation. Machine learning provides a powerful information processing framework that can enrich, and possibly even transform, current lines of fluid mechanics research and industrial applications.

1. INTRODUCTION

Fluid mechanics has long generated massive, heterogeneous datasets, while machine learning extracts patterns, automates tasks, and augments domain knowledge. This review surveys machine learning’s history, methods, applications, and limitations in understanding, modeling, optimizing, and controlling complex fluid flows.

  • Data and motivation: Experiments, field measurements, and large-scale numerical simulations have made massive data a longstanding reality in fluid mechanics.High-performance computing and advances in experimental measurement capabilities expanded fluid-mechanics data over the last decade.
  • Data and motivation: Machine learning extracts patterns and information from data, facilitating automation and augmenting human domain knowledge.The review frames these capabilities as relevant to fluid-mechanics research and applications.
  • Review scope: The review emphasizes understanding when learning algorithms succeed or fail and incorporating fluid-mechanics domain knowledge into them.It presents machine learning as an open and challenging field rather than focusing only on successful applications.
  • Historical development: Recent fluid-mechanics interest has grown with deep learning architectures that hierarchically extract informative features from data.The renewed blossoming of applications is associated with deep learning’s remarkable performance.
  • Fluid-specific challenges: Fluid dynamics requires precise physical interpretation because flows involve complex, multi-scale phenomena, nonlinearities, and unresolved challenges in understanding and control.Unsteady flow fields demand algorithms capable of addressing these complexities.
  • Fluid-specific challenges: Flow-control interventions can invalidate predictions from uncontrolled-system data, while fluid datasets may be sparse in some dimensions and highly heterogeneous.Parametric studies may be expensive, despite high spatial resolution in some fluid data.
  • Fluid-specific challenges: Machine-learning solutions for fluid dynamics should be interpretable, explainable, and generalizable, yet performance guarantees and convergence analyses remain rare.These requirements are especially important in transportation, health, and defense systems.

LEARNING FLUID MECHANICS: FROM LIVING ORGANISMS TO MACHINES

Biological organisms achieve remarkable fluid manipulation by optimizing shape and motion, while human fluid engineering has progressed from intuitive design to quantitative, physics-based methods. Machine learning frames fluid-mechanics inquiry as learning system outputs from inputs and observations, encompassing algorithms, flow modeling, optimization, and control.

  • Biological inspiration: Biological organisms manipulate unsteady fluid forces through optimized shape and motion for agile propulsion, efficient migration, and other maneuvers.Examples include birds, bats, insects, fish, and other aquatic and aerial lifeforms.
  • Human engineering: Humans are the only known species to manipulate fluids through knowledge of the Navier-Stokes equations.Human fluid engineering spans dams, irrigation, mills, and sailing, with performance advancing from intuitive design to quantitative, physics-based engineering.
  • Learning framework: A learning machine combines sample-generator inputs and system observations to approximate system outputs.The framework represents inputs as x, outputs as y, estimated outputs as ˆy, and machine parameters as w.
  • Review scope: The review covers fundamental machine-learning algorithms, flow modeling, optimization, and control before presenting a summary and outlook.These topics are organized across Sections 2–5.
  • Learning methods: Supervised learning uses labeled data and provides corrective information to the algorithm.

2. MACHINE LEARNING FUNDAMENTALS · 2.1. Supervised Learning

Machine learning estimates associations between system inputs, outputs, and parameters from limited observations by minimizing a risk functional. Supervised learning uses corrective information, especially labeled training data, while neural networks, recurrent architectures, and classification methods provide specialized approaches for nonlinear approximation, sequential data, and category prediction.

  • 2. MACHINE LEARNING FUNDAMENTALS: Learning machines estimate associations between inputs, outputs, and system parameters from limited observations through stochastic approximations.Their learning process is summarized as minimizing a risk functional.
  • 2. MACHINE LEARNING FUNDAMENTALS: The risk functional weights losses by p(x, y), which constrains predictive capabilities while balancing objectives such as accuracy, simplicity, and smoothness.
  • 2.1. Supervised Learning: Supervised learning provides corrective information through labeled training data, whose cost-function minimization determines the unknown parameters of the learning machine.
  • 2.1. Supervised Learning: The approximation function reflects prior knowledge, while linear-versus-nonlinear choices affect computational cost and alternative losses can impose constraints such as sparsity.
  • 2.1.1. Neural networks.: Neural networks use modular neuron-based architectures in which activation functions transform inputs, and layers can encode problem-specific structure and data characteristics.
  • 2.1.1. Neural networks.: Recurrent neural networks process ordered sequences such as video images and time series using weights learned by back-propagation through time.
  • 2.1.1. Neural networks.: LSTM algorithms mitigate diminishing or exploding gradients by using cell states and gating mechanisms to store and forget information about past inputs.
  • 2.1.2. Classification: Support vector machines and random forests.: Classification predicts labels or categories from a-priori labeled measurements by minimizing misclassification risk, with random forests offering interpretable, scalable tree ensembles.

2.2. Unsupervised Learning

Unsupervised learning extracts flow features without supervision or ground-truth labels through dimensionality reduction, quantization, and clustering. These representations support flow modeling and control, while dimensionality reduction can also preprocess supervised-learning tasks.

  • Unsupervised learning: Unsupervised learning extracts flow features using global criteria without supervision or ground-truth labels, including dimensionality reduction, quantization, and clustering.Automated feature extraction can support flow modeling and control through low-order models.
  • Dimensionality reduction: Dimensionality reduction maps high-dimensional flow data into lower-dimensional representations that filter information and can preprocess supervised-learning tasks.Extracting flow features from experiments and large-scale simulations is described as a cornerstone of flow modeling.
  • Dimensionality reduction: Proper Orthogonal Decomposition, or linear PCA, can be formulated as a two-layer linear autoencoder trained by stochastic gradient descent.This neural-network formulation provides an algorithmic alternative to linear eigenvalue/eigenvector problems.
  • Dimensionality reduction: Nonlinear autoencoders structure mappings between high-dimensional data and low-dimensional representations, potentially incorporating prior knowledge but risking loss of an inverse encoder–decoder relationship.The nonlinear encoder and decoder extend linear autoencoders.
  • Quantization and clustering: Vector quantization replaces a full dataset with representative points from a predetermined number of clusters, whereas clustering seeks meaningful groupings without that predetermined-center requirement.Vector quantization is data reduction rather than necessarily dimensionality reduction; k-means partitions data into k clusters using nearest centroids.

2.3. Semi-Supervised Learning

Semi-supervised learning operates with limited labeled data or corrective environmental information, including generative adversarial networks (GANs) and reinforcement learning (RL). GANs self-train competing networks after labeled initialization, while RL agents learn goal-directed behavior from experience-based rewards and face long-term credit assignment challenges.

  • Overview: Semi-supervised algorithms learn with limited labeled data or corrective information from the environment, including GANs and RL.Both approaches use a game-like process in which the learning machine is (self-)trained.
  • Generative Adversarial Networks: GANs generate candidate data while discriminators evaluate it, with both networks competing in a zero-sum game to optimize the task.The generator produces data intended to mimic the training-data distribution, while the discriminator distinguishes true from generated examples.
  • Generative Adversarial Networks: GAN networks essentially train themselves after initialization with labeled training data, with the discriminator maximizing and generator minimizing discrimination probability.The minimized function is the Kullback-Leibler divergence between the two distributions.
  • Reinforcement Learning: RL agents interact goal-directedly with environments, selecting actions from perceived states and learning from rewards that may be infrequent and partial.Unlike supervised learning, RL provides no labeled information about the correct actions.
  • Reinforcement Learning: Tesauro’s backgammon learner trained by playing a couple of million games against itself, won the computer backgammon olympiad, and became comparable to the world’s three best human players.The program began from scratch as a novice player.
  • Reinforcement Learning: A core RL challenge is long-term credit assignment, which requires inferring causal links between individual decisions and rewards across long sequences, especially with sparse or delayed rewards.Efforts address this by augmenting sparsely rewarded objectives with denser rewards and accounting for past experiences.

2.4. Stochastic Optimization: A Learning Algorithms Perspective

Stochastic optimization can be viewed as learning the probability distribution of design points that maximize an objective, linking optimization and search algorithms with learning algorithms. Evolution strategies and genetic algorithms have increasingly converged with estimation-of-distribution algorithms, exemplified by CMA-ES.

  • Stochastic Optimization: A Learning Algorithms Perspective: Optimization can be formulated as learning the probability distribution of design points that maximize a specified objective.This perspective extends the role of optimization beyond minimizing a learning risk functional to identifying promising design points.
  • Stochastic Optimization: A Learning Algorithms Perspective: CMA-ES adaptively estimates a Gaussian distribution’s covariance matrix to guide parameter search, updating it iteratively from the best points in each iteration.The algorithm is a prominent evolution-strategy example within the estimation-of-distribution framework.

2.5. Important Topics We Have Not Covered: Bayesian Inference, Gaussian Processes

The review highlights Bayesian inference as an important unaddressed learning approach for fluid mechanics. It frames Bayesian methods as probabilistic, data-driven tools for informing model structures and parameters, quantifying uncertainty, and connecting machine learning with optimization.

  • Bayesian inference uses data in a probabilistic framework to inform model structures and parameters.
  • Bayesian inference is fundamental for uncertainty quantification because data adapt model estimates.
  • Machine-learning algorithms can alternatively be formulated within a Bayesian framework, with optimization algorithms linking the two perspectives.

3. FLOW MODELING WITH MACHINE LEARNING

Machine learning complements first-principles flow modeling by extracting reduced representations and learning flow dynamics from increasingly data-rich experiments and simulations. The review distinguishes modeling flow kinematics through feature extraction from modeling dynamics through learning architectures, while emphasizing data requirements and physical constraints.

  • Motivation: High-Reynolds-number scale-resolving Navier–Stokes simulations exceed current computational resources, motivating approximate simulations, experiments, and data-driven alternatives.First-principles conservation laws remain foundational, but computational limits make alternative modeling approaches necessary.
  • Modeling directions: Dimensionality reduction extracts dominant flow features as reduced coordinates, whereas reduced-order modeling describes spatiotemporal evolution through parametrized dynamical systems or statistical maps.These are complementary directions for compactly and efficiently representing fluid flows.
  • Modeling directions: The review separates machine-learning approaches that model flow kinematics through feature extraction from those that model dynamics through learning architectures.This distinction organizes the subsequent discussion of machine-learning methods for fluid modeling.
  • Methods: POD, PCA, autoencoders, deep neural networks, clustering, and classification provide tools for reduced coordinates, latent representations, phase-space discretization, and regime identification.POD supplies an orthogonal empirical basis, while autoencoders compress data into low-dimensional latent spaces and clustering can yield tractable Markov transition models.
  • Measurement and computation: Super resolution infers high-resolution flow fields from low-resolution measurements by leveraging statistical structure in high-resolution training data, including particle-image-velocimetry data.Approaches include example libraries, sparse representations, and convolutional neural networks.
  • Limitations and challenges: Neural-network models are fundamentally interpolative, so extrapolation requires caution, especially for high-dimensional unsteady flows with multiscale dynamics, noise, latent variables, and transients.Fluid-dynamics learning must also address partial prior knowledge, governing-equation constraints, and symmetries.

4. FLOW OPTIMIZATION AND CONTROL USING MACHINE LEARNING

Machine learning supports flow optimization and control, especially for black-box or multimodal cost functions, but learning-based methods can require many evaluations and lack convergence guarantees. Stochastic optimization has broad engineering applications, while real-time flow-control optimization, explainability, and reliability remain challenging.

  • Learning-based optimization and control: Learning algorithms are suited to flow optimization and control problems with black-box or multimodal cost functions.They are iterative and may require several orders of magnitude more cost-function evaluations than gradient-based algorithms.
  • Learning-based optimization and control: When adjoint methods are applicable, learning algorithms are suggested against because they lack convergence guarantees and can be evaluation-intensive.The passage recommends avoiding learning algorithms in such cases.
  • Stochastic optimization: Stochastic optimization has been applied to aerodynamic shape optimization, UAVs, artificial swimmers, and power extraction in crossflow turbines.These applications illustrate its widespread use in engineering design involving black-box cost functions.
  • Stochastic optimization: Stochastic optimization algorithms use large numbers of iterations, benefit from massively parallel architectures, and have been applied in experimental and industrial settings.They are well-suited to challenges involving uncertainty, including unexpected system behavior and partial descriptions of the system.
  • Challenges and future directions: Convergence proofs, explainability, and reliability remain outstanding issues for stochastic flow optimization, while hybrid stochastic and gradient-based methods may be effective for flow control.Real-time optimization is also challenged by the critical spatiotemporal scales and nonlinearity of flow processes.

OPTIMIZATION AND CONTROL: BOUNDARIES ERASED BY FAST COMPUTERS

The section presents optimization and feedback control as increasingly intertwined as faster computers enable optimization to be solved on the fly as feedback laws. It reviews machine-learning approaches for flow control, system identification, sensing, and actuation, emphasizing both their capabilities and practical challenges.

  • Optimization and control: Faster computers increasingly erase the practical boundary between optimization and feedback control.When optimization can be solved fast enough on the fly to close the loop, it can constitute a feedback control law; the same algorithm may be viewed as optimization or control depending on hardware capabilities.
  • Feedback flow control: Feedback flow control uses sensor-informed actuation to modify fluid dynamics while addressing instability, noise, disturbances, uncertainty, and difficult system structure.Flow-control challenges include high-dimensional states, nonlinearity, latent variables, and time delays; machine learning has been used for control, system identification, and sensor placement.
  • Neural-network control: Neural networks learned sensor-based control laws for turbulent boundary layers, reducing skin-friction drag with local wall-normal blowing and suction from few skin-friction sensors.The learned law was based on a known optimal full-information controller and incurred little loss.
  • Evolutionary methods: Genetic algorithms and genetic programming have addressed flow-control design, including experimental mixing optimization, multi-frequency actuation, sensor feedback, and distributed control.Genetic algorithms require a pre-specified control-law structure with only a few adjustable parameters, while learning time increases with the number of parameters; most genetic-programming control laws were obtained within 1000.
  • Reinforcement learning: Reinforcement learning applies agent-environment interaction to fluid problems by learning policies that maximize long-term rewards.Successful applications depend on selecting states, actions, and rewards that reflect flow mechanisms, with examples spanning bluff-body control, collective swimming, gliders, UAVs, and microswimmers.

5. DISCUSSION AND OUTLOOK

Machine learning can augment fluid-mechanics modeling, optimization, control, and scientific inquiry by combining data-driven methods with domain knowledge. Its benefits depend on careful problem formulation, method selection, training, validation, and integration of known physics.

  • Contributions: Machine learning provides tools for modeling, optimizing, and controlling fluid flows in experiments and simulations, including dimensionality-related fluid-mechanics tasks.The review highlights applications across flow modeling, optimization, and control.
  • Contributions: Fluid-mechanics principles, including conservation laws, can frame precise questions and reduce computational costs in machine-learning-based flow control and optimization.Domain knowledge also remains relevant in the era of big data.
  • Limitations and challenges: Machine-learning answers depend on training data and require careful question formulation, data selection, algorithm choice, training, cross-validation, and attention to interpretability and extrapolation.Method selection should also consider data quality and quantity, desired inputs and outputs, cost functions, and whether explainability is important.
  • Outlook: Combining first-principles theory with data-driven analysis could address long-sought fluid-dynamics problems and improve understanding of turbulence and its governing mechanisms.The paper characterizes this combination as a new era in fluid-mechanics research.
  • Outlook: Effective fluid-mechanics problem solvers need expertise in both machine learning and fluid mechanics, as machine-learning capabilities continue advancing rapidly.Fluid mechanics has used machine learning for decades to understand, predict, optimize, and control flows.

DISCLOSURE STATEMENT

The authors report no known affiliations, memberships, funding, or financial holdings that might be perceived as affecting the review’s objectivity.

  • The authors are unaware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the review’s objectivity.

The classic book for reinforcement learning

The section identifies Sutton and Barto’s 2018 Reinforcement learning: An introduction as a foundational reinforcement-learning book. It specifies the second edition and MIT Press as its publication details.

  • Sutton RS and Barto AG authored the cited reinforcement-learning book in 2018.
  • The book is titled Reinforcement learning: An introduction.
  • The citation identifies the work as the second edition, published by MIT Press.
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